%C Robert Castelo (rcastelo(AT)imim.es), Jan 06 2001, observes that n^(n-2) is also the number of transitive subtree acyclic digraphs on n-1 vertices.

%C a(n) is also the number of ways of expressing an n-cycle in the symmetric group S_n as a product of n-1 transpositions, see example. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 12 2001

%C The parking functions of length n can be described as all permutations of all words [d(1),d(2), ..., d(n)] where 1 <= d(k) <= k; see example. There are (n+1)^(n-1) = a(n+1) parking functions of length n. - _Joerg Arndt_, Jul 15 2014

%C a(n+1) = number of endofunctions with no cycles of length > 1; number of forests of rooted labeled trees on n vertices. - _Mitch Harris_, Jul 06 2006

%C a(n) is also the number of nilpotent partial bijections (of an n-element set). Equivalently, the number of nilpotents in the partial symmetric semigroup, P sub n. - _Abdullahi Umar_, Aug 25 2008

%C a(n) is also the number of edge-labeled rooted trees on n nodes. - _Nikos Apostolakis_, Nov 30 2008

%C a(n) is the number of acyclic functions from {1,2,...,n-1} to {1,2,...,n}. An acyclic function f satisfies the following property: for any x in the domain, there exists a positive integer k such that (f^k)(x) is not in the domain. Note that f^k denotes the k-fold composition of f with itself, e.g., (f^2)(x)=f(f(x)). - _Dennis P. Walsh_, Mar 02 2011

%C a(n) is the absolute value of the discriminant of the polynomial x^{n-1}+...+x+1. More precisely, a(n) = (-1)^{(n-1)(n-2)/2} times the discriminant. - _Zach Teitler_, Jan 28 2014

%C For n>2, a(n+2) is the number of nodes in the canonical automaton for the affine Weyl group of type A_n. - _Tom Edgar_, May 12 2016

%F For n>=3 and 2<=k<=n-1, the number of trees on n vertices with exactly k leaves is binom(n,k)S(n-2,n-k)(n-k)! where S(a,b) is the Stirling number of the second kind. Therefore a(n) = Sum_{k=2}^{n-1}binom(n,k)S(n-2,n-k)(n-k)! for n>=3. - _Jonathan Noel_, May 05 2017